The MST won't change, because the relative order of edge weights stays the same. Therefore, by Kruskal's algorithm, the MST will stay the same. Another way to see this is the following: Each spanning tree has n-1 edges. So by increasing each edge weight by 1, the weight of each spanning tree increases by n-1. Would the MST change, the original spanning tree wouldn't have been minimal.

The shortest path can change, however: Consider a graph with 4 vertices and edges (1, 2), (2, 3), (3, 4) of weight 1 and edge (1, 4) of weight 4. Before increasing the edge weights the shortest path from 1 to 4 has length 3 (1-2-3-4). After the increase, the length of this path becomes 6. This is no longer the shortest path, because simply using the edge (1, 4) has length 5. Compared to the spanning trees this issue arrises precisely because shortest paths don't have a fixed number of edges.